TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE IN A LAYERED MEDIUM

TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE IN A LAYERED MEDIUM *

BY

T. LEE **

ABSTRACT LEE, T., 1975, Transient Electromagnetic Response of a Sphere in a Layered Medium,

Geophysical Prospecting 23, 492-512.

The Galerkin method of solving integral equations is well suited to the solution of the integral equations describing the transient response of a sphere embedded in a layered medium, which is excited by a large co-axial loop.

The transient response is calculated by transforming the steady state solutions obtained in the frequency domain.

The analysis shows that the scattering matrix is extremely diagonally dominant and the maximum number of modes required to obtain convergence does not rapidly increase with frequency. The number of modes required is about eight. This type of scattering matrix can be taken to be an expression of the principle of elementary superposition. This principle is reflected in the decay curves. These show that the early part of the decay curves asymptotically approach the decay curves to be expected for a layered structure without the sphere. The slope of the latter stages of the decay curve gives a decay constant that is the same as was obtained for spheres in free space excited by planar or dipolar sources.

The point of departure in time of these curves from the layered ground curves is delayed either by placing the sphere at a greater depth or by placing a more conductive overburden above the sphere.

I. INTRODUCTION

The transient electromagnetic prospecting system employing a single large loop is at present in widespread use in the mineral mining exploration industry. Unfortunately, however, there are very few geometries for which the transient response has been modelled. This is particularly so with regard to numerical modelling, and consequently the most useful results which can be used for the interpretation of field results are the analogue model results of Velekin and Bulgakov (1967). To date the only numerical results which have been available are those given by Bulgakov (1971) for a uniform ground, Kamenetski (1968)

* Received September 1974. * * Formerly School of Earth Sciences, Macquarie University, North Ryde 2113 NSW

Australia; now at L. A. Richardson and Associates, 33 Bertram Street, Chatswood, NSW, 2067 Australia.

TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE 493

for the response of a thin conducting layer above a uniform half space and the results of Lee and Lewis (1974) for a multi-layered ground. Further, non layered structures have only been considered under very restrictive conditions.

One particular model which has received considerable attention has been a spherical conductor. While it has proved possible to use a sphere composed of a number of layers the resulting theoretical models were still very restrictive since the sphere was considered to be placed in an infinite uniform medium and excited by either planar or dipolar fields (Nabighian 1970, Verma and Singh 1969, Singh 1972, and Mallick 1972). Thus, Nabighian (1970) considered the quasistatic response of the multiples induced in a conducting sphere by a dipolar field which was abruptly removed at t = o and found that, for large values of time, the field decays purely exponentially with a decay constant of x~/(cs@) where 6, b, and p are the conductivity radius and permeability of the sphere, respectively.

Subsequently, Verma (1972) showed that the transient response for a uniformly conducting non-permeable sphere to a variety of different pulses could be rapidly obtained by the use of the convolution theorem for the Laplace transform.

In 1972 Negi and Verma tried to model theoretically Bulgakov’s and Velekin’s (1967) modelling experiment in which a conductive sphere is placed below a thin conductive sheet. The result was that the earlier stages of the transient response of the system corresponded to the sheet alone while the later stages corresponded to the sphere alone. The approach used by Negi and Verma (1972) was to excite two concentric spherical shells of different thick- nesses, placed in a uniform space by a uniform magnetic field. That the above type of response could be expected had previously been indicated by the results of Negi (rg67), Nabighian (1971) and Fuller (1971).

In 1967 Negi studied a two layer sphere geometry and found that the electromagnetic response of this structure was, for some frequencies larger than if the inner sphere was not present at all. This then led to the concept of negative screening. Subsequently, Fuller (1971) also considered a two layered sphere and showed that the core and outer shell in general respond in a different manner for a given frequency. That is, if the conductivity of the shell is greater than that of the core the predominant contribution is made by the shell. For the case where the shell conductivity is lower than that of the core the shell and the core responses-separate in frequency-are not greatly effected by each other, the core and the shell responding to lower and higher frequency excitation, respectively.

While the usual approach to the study of the transient response of a spherical conductor is via the Laplace transform of the steady state solution, Hjelt (1971) showed how to calculate the transient response of a two layered sphere

494 T.LEE

by assuming that the decaying field is represented by the sum of a set of decaying experimental fields. This analysis was carried out for a homogeneous field disappearing at t = o. One conclusion from this study was that the decaying field could be represented by a single decay constant when +t/

(cpR2) > 0.1. Here tr, p”, and R are the conductivity, permeability, and the radius of the sphere, respectively.

In spite of the above analysis very little is known about the exponentially decaying response to be expected from a conductor which is buried within a half space. Wait and Hill (1972) have partially considered this problem by analyzing the fields observed at the surface of a half space, which were due to a transient loop source buried within the half space. The results given by Wait and Hill (1972) show that the decaying voltage induced in a receiving loop depends on the ratio of the horizontal distance to the axis of the loop and the depth to the loop from the surface to such an extent that, in the opinion of these authors, it is possible to use the wave form as a readily detected criterion of source location for a simple observation point on the earth’s surface.

Singh (1973) made a study of the transient electromagnetic response of a spherical conductor in a conducting full space. He concluded that for the case of the sphere being excited by an arbitrary oriented dipole source it was not always possible to neglect the conductivity of the host material. Specifically, Singh found that the conductivity of the host material could not be neglected if the following conditions were true.

(I) I/IO0 '< m/o2 < I/IO, (2) PZ/Pl = 1, and

61 lo (3) (01/4 i > 0.2 and O,

0 5 > o.2,

In these conditions ~1 and ~1 are the conductivity and permeability of the host material while 02 and‘y2 are the corresponding quantities for the spherical conductor. The radius of the sphere is a, while vo and Y are distances from the center of the sphere to the exciting dipole and observation point, respectively.

The analysis to be presented below can be extended to a much more compli- cated geometry than any of the above cases. In particular, it will be shown how the following analysis can be used to allow for the finite dimension of the source and yet allow for a number of layers to be placed above the half space in which a conducting sphere is embedded.

2. FORMULATION

The formulation of the problem, to be given below, is a particular case of a more general method for solving integral equations. This “method of moments” has found wide use in electrical engineering where it has been used very suc-


cessfully. For an account of the general method the reader is referred to Harrington (1968).

For the particular problem to be considered here we shall represent the electric field by the spherical wave functions. As these functions are orthogonal over the sphere the moment method can be reduced to the Galerkin method for the solution of integral equations.

Since in what follows there is a considerable amount of tedious algebra the method is symbolically outlined below.

Consider the linear Fredholm Integral equation :

g(x) - S K(x, Y) g(y) dy = f(x) where x E D, D

where

D is a compact subset of R”, v 3 I.

(1)

Let g(x) = $ g&&) b e a complete representation for g(x) in the domain *=1

D and choose $i, 4~ SO that J $& dy = 8~. D

Further, let f(x) = Ecf~#c (x)

and K(% Y) = c)= KZi$&) h(Y) = c 4(x) MY).

Insertion of these expressions into the integral equation (I) and using the orthogonality property of the functions 4 yields

2 g&c - XX &j $6 gj = ~fiqW). (2)

When equation (2) is multiplied by & and integrated over the domain D the following system of equations results:

gi - C Ku gj = ft, (3)

If the series in equation (3) is terminated after N terms, there results a system of equations from which estimates of gi = & say, can be determined.

Consequently

We shall not discuss the convergence of such a solution as a function of N but simply state that the method has been found to yield excellent results. A review of the method has been given by Ikebe (1972).

496 T. LEE

Clearly, if such a method, as advocated above, is to be an economic method of solving scattering problems, then a series of very difficult integrals must be evaluated. The main results.needed, however, are shown in table I, which also shows the sources of the formulae or their method of derivation.

TABLE I

Some formulae and their derivation

[II (W - h2) S zBy(k,z) bdkzz) dz = x [k,W hz) Wwz) - k,BdW W VW1 = - x Wy (k,z) By+1 (W - ktBdkl4 by+1 bW1

where B,(x) and by(s) are Bessel Functions of order y. Luke (rg6z), p. 254

(1)

(2)

where JY(z) and H,r(z) are Bessel and Hankel Functions of the first kind, respectively. Magnus, Oberbettinger, and Soni (1966) p. 68

131 &ZCOSfJ = v-

$ *;@ + I) i” Jn+l/,(z) P, (cos 0) (3)

Watson (1968) p. 368

[41 etkR in m -= - 2 LEO '7; I in +!/z (uk) f&+1/, (VI Pn @OS -d C-

R (4)

where G( < p, R = Vu2 + 8” - zc+ cos y

Watson (1968) p. 366

* (n + k)! 2 = rP,k (cos e) P,k (cos e) sin 8~03 yT=q--k)! 0

Macrobert (1967) p. 117 N.B. P,m (COS y) = (I - COS20)m/2 dm/d COS em pn (COS e)

[5J

(n - I)! P,I (~0~ +) P,’ tcos e) ___

(n + I)!

where nr = 1/m, and v-. This is obtained from (3) by writing k sin J, = h and then using the integral representation for the Bessel function given in (7) below and finally the addition theorem for the Legendre functions. Macrobert (1967) p. 127


[71 (7)

Magnus et al (1966) p. 7g

ix ma (n-1)! (292 + I) -

."I (n + I)! yap ____ Jn +‘/z (ah) Hn+1/, (pk) P,l (cos 0) P,’ (cos 01) (8)

2

where nl = I/hz-kz apd p < CL.

This result is obtained by using the addition theorem of order zero for Bessel functions (Magnus, Oberhettinger, and Soni 1966, p. 107) to obtain a representation for the product of two Bessel functions as an integral involving the zero order Bessel function. The order of integration is then interchanged and Sommerfield’s result is used to evaluate the inner integral.

Finally, the expression is expanded with the aid of equation (4) and the integration is performed term by term, (cc, 0) and (p, 01) are the coordinates of the points (a, z) and (b, z) respectively if

l/b2 + .i? > 1/a2 + 9.

Otherwise the relations are interchanged.

The special geometry which we shall consider is shown in fig. I.

This diagram shows a sphere of radius b and conductivity 1s2 embedded in a half space of conductivity ~1. Above the sphere and co-axial with it is a large loop of radius a. For this analysis we shall assume an egwt time dependence, neglect displacement currents, and set the permeability Al. equal to the free space permeability everywhere.

Fig.-I. Sphere of radius b and conductivity 02 embedded in a uniform halfspace of conductivity ts~. A large loop of radius a lies on the ground surface.

498 T.LEE

The electric field E, of which there is only a horizontal component, therefore satisfies the following integral equation (Hohmann 1971)

E(x) = I?(,) - Pi - k:)

zxiwp s E(a) G@, ~1) d;. (4)

Sphere

In equation (4), - kj = iwpoj ; E” is the incident electric field which is defined to be the electric field which would exist within the medium if there were no heterogeneity present. The Green’s function G(z, ~1) is simply the axial symmetric analogue of the Green’s function considered by Hohmann (1971), and is therefore just the electric field due to a ring source of current in the earth. Consequently, it can be readily derived from the expressions given by Morrison, Phillips, and O’Brien (1969) for the field of a ring source in a layered medium. Hence

m

G = - ?$ J‘ y1 J1 ($A) J1 0,~) e-R kzl) + (zi) emnl(lxl)] i d?,

0 (5)

The electric field E” if found in an analogous manner to be:

Eg (Y, z) = - iw&r s

J4W J1 (WA e- n,z dh n1 + A (5.5)

Here I is the strength of the current in the loop of radius a. In equations (4, 5) Jo are Bessel functions of argument z and nl = 1/h2 - k2.

In order, to solve the integral equation by the method outlined above we require a complete representation for the electric field E over the domain bounded by the sphere.

For this purpose we set m

E= c

J s + ‘/z (W Pia (~0s 0) An 9&=1 G

(6) .

where the coefficients A, are as yet undertermined.

This is merely the solution which would be obtained for the electric field if the wave equation in spherical polar coordinates were solved for the electric field within the sphere. By using result 8 in table I the Green’s function can be simplified to

s (y, yl, / z - .zl 1 ) + s JI (1~) Jl (AC) (z;) ; e-“‘(z+zl) dh] (7) 0


where . m

SI(T,Y~, Iz--11) = ; c zrt+I Jn+y,(h~)~~+,/z(P~~)~ n-I G

P; (COS e) P; (COS e,) (n - I)!

(a + I)! (8)

All symbols have their previous meaning.

The second portion of the Green’s function is simplfied by writing

G= S1(r, r1, I z---1 I ) 3

a[ $ (z) Jl(Ar) J~@rl) e~~(h-zl)-fll(z+h~) ah] (9)

where h is the depth to the center of the sphere.

The subscripted terms (which are the terms over which we integrate in the Green’s function) can be transformed to the spherical polar co-ordinate system, which describes the sphere’s surface, by using result number 6 of table I.

Therefore,

Looking now at equations (4) (5) (6) and ( IO we see that the integrations ) over, the sphere, involved in equation (4) can be done directly, since the orthogonality conditions of the Legendre functions, result 5 of table I, can be used to effect the 0 integrations, and then the integrations of the resultant product of the Bessel functions can be performed by using the results I and 2 of table I. The integration of the first term in the Green’s function is performed by splitting the integral into two parts, the first from o to R and the second from R to b, with the appropriate expression for S being used in each case. The resulting integrations are simplified by the use of result 2 of table I. In the second term the integration with respect to RI is done directly. Geophysical Prospecting, Vol. 23 32

500 T.LEE

Therefore, equation (4) has been reduced to

E(R1, e) = Et(R, e)+ 2 An Jfi+1/, (~9 Pi (~0s 0) IF

- P; (cos 0) ixb. 11=-l

J .AnWn

n+l/z (W) ___ 2 E (Jn+)/z @2% H;+l/z(W) - x - - zkl

in-lb W, (Jn+% (M)> Jn+% (W) An *

(11) cl

where

Wn (Bn+,/z (k24, b,+s (hb))

= kl B, + 1,2 @A bn + s/2 @lb) - k,B, + s/2 (hb) b, + 1,2 (hb).

The above corresponds to equation (2) which was given previously in the symbolic solution to the problem.

The next logical step is to multiply equation (II), throughout by P& (cos f3) sin 0 and to integrate over the surface of the sphere.

The resulting integrations are performed as follows :

Firstly, both the incident electric field and the integral term in equation (II) are expanded in a series of spherical wave functions as was done previously by using result number 6 of table I. Then the integrations are readily performed by interchanging the order of integration and summation and integrating term by term.

The results are

- 20 I*Ia

= =

c

inb J, +% (k&‘) (n + I) !

ll-I vi?- (S-II)!

I---

A&nn ' z Wn (Jn+l/,(kzb), Hi+y, (bb))

inwlbWn (J,+t/, (k&), Jn+l/z (hb)) An * Jm+1/, (kl')*

. J pi (!Z) pk (!!) i . (zi) e-ra,; dh (12) 0


Truncating after N terms we obtain the following system of equations which yield estimates A, for the A,

N I - c -i-nxb - I Zkl (fi + I)! An&u72 = pWnWn+,

x (n-1)! 2?2+ I &A, ~~+~/z(W) +

m-1

+ zi-“bWn(Jn+1/2 Chb),Jn+1/2 (‘lb)) Ati

(13)

Equations (13) can now be solved to yield a set of coefficients A, which are estimates for the A;.

Thus, if the An’s are assumed to be known they can be inserted into equation (4) to yield an expression for the electric field outside the sphere.

. N -

:. E = Ei -‘L” c

A, Hl,+% (k,R)BP; (cost)). 2 n-1 E

Wn (Jn+1/, (k~b)aJn+, (hb)) - ” P-l b/f, W,(J zkl

n+s(bb)*Jn+1/, (hb)).

(14)

To see how the above method can be employed to solve equation (4) when there are a number of layers above the sphere we shall consider briefly the case where the sphere is embedded in a two layered medium.

The sphere is assumed to be in the lowermost layer. Such a model can be used as a means of studying the effect of a conducting

overburden, which is a very real situation in many parts of Australia. For a layer of thickness d and conductivity 61 above a half space of con-

ductivity csz in which there is a sphere of radius b, conductivity bg and depth to the center h, the integral equation which is to be solved is:

k”3 - k2 (15)

502 T.LEE

In this equation k2 = - iwp~al when the field is evaluated in the upper layer, and ----iti pas when the field is evaluated in the lower medium (Hohmann 1971).

In what follows ki = - ioyaj and tij = 1/~~ - ki. For this model the Green’s functions, which can be readily found from the

expressions given by Morrison et al (rg6g), are

GE- y j rlJl (A#)J (A~) [e-lZ-Z1i%+ e-%(Z+Z1-2d)

b2 - n,)l(h'+ 4 + (75, + f-b) (n, - A) e-2nlal

(n2 + 4 h + A) + (n2 -n,) (lzl - h) ewznld 11 _1. dh

n, (16) for the lower medium, and

Jl(hrl) J,(k) 4 ~5~~2~ emnnz dh f-2, ((A + 92,) (f-2, + n2) e(nl-nz)d + (h-q) (a1 - n2)e:(nl+n2)d) o

(17) for the upper layer.

The incident electric field is described by

Et= -iiwpa -A

J- J1(ha) J,(k) z~,n, emnzz dh

nZ [(A++4 h+fl,) e m-ne)d + (A--~) (1~~ -fi2) e-(nl+na)dh] 0

(18) The new set of simultaneous equations can now be readily found by compar-

ing equations (16) and (18) with equations (5) and (5.5). Therefore the analogue to equation (13) is

* A -2ziwpaI - -

s Jr(ha) zn,n, e- nzh Pk (in,/k,) dh

nz ((h+n,)(rt,+fi,) e+(nl-nz)d+ (A-9~) (n,---~2~)e-(~~+~~)~) 0

= ,f - ci)-” xb 1/” iz + Ei: s * Wn (Jn+1/, (ksb), Hi+% (k,b)) x (1-1

+ G-” b J+‘n (Jn+1/, (kd), Jn+s (klb))‘An s1 (-I)?

TRANSIENTELECTROMAGNETIC RESPONSE OF A SPHERE 503

The electric field at the surface of the ground and particularly at the loop is found by using equations (IS), (17), and (18) together with the now assumed A,. Therefore :

4w2 e - n2z1 &,

(A+n,) (n1+n2)e(nl-n2)d+(h--n,) (n,--+z2)e-(n1+n2)d ’

m Jn+4/2 c (k2R) pi Ccos e, An dv (20)

n-1 E

By using result 6 of table I the above expression readily simplifies to

E = Et - b’ (k’ - “) 2 A, (k, Jznml (k,b) Jn+3,2 (k,b) - k; - k; n-1

-bJn+l\, (bb) Jn+3/2 (hb)) in-l ” * zkz

-?i s e - nzh Pi (in2/k2) 4n,n, d h -- 922 [(A + n,) (nl + n2) e(nl-na)d + (h-nl) (n, -9~~) e-(nl+na)al] (‘I) cl

3. NUMERICAL RESULTS AND DISCUSSION

Since the voltage induced in the loop (fig. 2) by a step current excitation of a sphere embedded in a layered space is axaE, where E is the electric field evaluated at the loop, it can be readily obtained from the equations given above. This, then, yields the steady state solution for this particular problem, and the transient solution can be obtained by the inverse Laplace transform of these solutions. The procedures employed to obtain both the transient and steady state solutions are described by Lee (1974).

(a) Steady State Solution

It is instructive to examine the response in the loop of a buried sphere for the steady state situation where a current of Ieiwt flows in the loop. Such an ’ analysis, as will be shown, enables us to examine the relative importance of the various terms in the above equations and to gauge the strength of the inter- action between the ground air interface and the spherical conductor as is expressed in the off-diagonal elements of the scattering matrix.

504 T. LEE

Fig. 2. The coordinate systems which are employed to describe the fields. The loop of radius a is described by a cylindrical coordinate system Y, 4, z. The sphere of radius b and

center z = h is described by a spherical polar coordinate system R, 0, $.

TABLE z

Ratio of the off--diagonal element in the motion (OD) to the diagonal element (D) fey a variety of depths and frequencies. NM is the number of modes reqzlired to

secure convergence of the solution.

Radians/ Sec.

kl kz a b h NM Approximate Ratio 00/D

I.96 x 105 0.0496 0.496 50m 50m 5om 7 10-S I.96 x 105 0.0496 0.496 gom gom room 6 10-4 1.25 x 103 3.97 x IO-3 3.97 x 10-s 5om gem 5om 4 10-4 1.25 x 103 3.97 x IO-3 3.97 X IoQ 50m 50m Ioom 4 10-4

This analysis is particularly instructive in the light of Howard’s (1972) results for a line source exciting an infinite cylinder of radius Y embedded in a half space. Howard found that for “electrically small” cylinders- 1 ka 1 < I

but with no restriction on 1 kzr 1 -only a few terms in the multipole field expressions were required, and that the corresponding scattering matrix was extremely diagonally dominant and consequently very well conditioned numerically. The numerical results given in table z below show that this is also true for “electrically small” spheres i.e. when 1 klb 1 < I. This table shows the ratio of the off-diagonal elements of the scattering matrix to the diagonal


elements for a variety of depths and frequencies. Also shown is the number of modes NM which were necessary to secure convergence of the series. Further discussion on the convergence of the series is given by Lee, (1974).

Consequently, for this model the off-diagonal elements can be neglected in order to obtain an accuracy of 1%. Further, the number of modes which is necessary to obtain a solution is quite small and does not rapidly increase with the electrical size of the sphere or its depth of burial. Even when a conductive overburden was placed above the half space the number of modes remained small (less than 7) and again the off-diagonal elements of the matrix were smaller than the diagonal elements although their effect was more pronounced.

A particularly attractive feature of the mode-matched solutions is that the number of modes does not increase very rapidly with frequency, and consequently it is ideally suited for the treatment of transient problems. The one difficulty experienced was that for large frequencies, (o > 10~ radian/s.) some care had to be exercised with the numerical evaluation of the integrals, since the phase of the function being integrated changes rapidly for these frequencies.

Although only simple geometries have been treated here, the method is capable of considerable extension. Thus, Andreasen (1965) considered the problem of scattering of a plane electromagnetic wave from a conducting body to expanding incident wave into cylindrical modes. Andreasen points out that when conventional methods are used the maximum size of the body which can be treated economically is about one wavelength, while for the mode matching method bodies of the size of twenty wavelengths can be handled.

More recently, Garbacz and Turpin (1971) and Harrington and Mautz (1971 a, b) have extended the. mode matching method to arbitrarily shaped bodies.

The approach used by these authors to obtain solutions of Helmholtz’s equation for surfaces for which it is not separable is to diagonalize either the scattering matrix or the operator relating the current to the tangential electric field on the body. The conclusion reached by these authors is that only a small number of modes is required to characterize the electromagnetic be- haviour for electrically small or intermediate size bodies.

(b) Transient Sohtion The above method of obtaining steady state solutions to the electromagnetic

scattering problems has been used to calculate the quasistatic response of a buried sphere for a variety of depths and loop sizes (figures 3-7). The conse- quences of this simpler analysis resulting from the neglect of displacement currents will be discussed in another paper. Some experimental verification of the results to be presented below will be given in that paper also.

506 T.LEE

The modelling experiment only verified the computer programs used here for signals greater than 10.0~~ volts per amp. and for this reason signals less

IO-’ -

IO” -

than this are not generally given.

F4

elOOm--

(I I Uniform ground q= 0.01 (2) h=75m qe2.0 (3) h= IOOm q2.0 (4) h= 150m $=2,0

10-s ’ I I I’ I I , I I I 0 0.5 I.0 I.5 . 2.0 2.5 3.0 3.5 4-o 4.5

f -millisecondr

Fig. 3. Transient response of a coeducting sphere in a half spice for various depths to the center of the sphere. Loop larger than sphere.

Fig. 3 shows the transient response of a sphere embedded in a half space for various depths to the center of the sphere. These curves show that the early part of the curves asymptotically approach the transient response of a uniform half space. The effect of depth, then, is slowly to decrease the amplitude of the signal and to delay the departure of the curves from the uniform half space response.

Fig. 4 shows the effect of varying the loop size on the decaying waveform. The results show that the effect of increasing the loop size is to increase the


magnitude of the voltage at all times and indicate that the spherical target is best excited by a large loop. It is suggested that these latter curves reflect the strong coupling between the currents in the sphere and the conducting environment for these early times. Consequently, when carrying out field surveys it is advisable to use larger and larger loops so as to obtain a large enough

F 4.~

a Varies

h = 75m

li-

50m U’O~OI a=2,0

\ I , l”“lll a”“* YllllVllll “9”.

(21 IOOm loop with sphere (3) 50 m loop with uniform ground (4) 50 m loop with sphere (5) 30 m loop with uniform pround (6) 30 m loop with sphere

4

t - milllreconds

Fig. 4. Transient response-of a conducting sphere in a half space for loops of various sizes.

signal for the latter stages of the transient signal which, as will be shown below, are apparently controlled largely by the spherical conductor.

As stated in the introduction to this paper, previous authors had found that for large time the electric field about a conducting sphere in free space decayed purely exponentially. This was true regardless of whether the sphere was excited by either planar or dipolar sources. Further, Velekin (1971) showed

508 T.LEE

that this was also true when the spherical conductor was excited by a step current flowing in a loop of wire which is co-axial with the sphere. The principal decay constant, then, was defined to be +/(cpb2) where 0, p and b are the conductivity permeability and radius of the sphere, respectively.

,#’ Volt$/Amp F 4.

-5Om-h

II) Uniform ground 12) h= 70m. (31 h = 75m. (4) h = IOOm.

IdSo I I I I I I I I I

0.5 I.0 I.5 2.0 25 3.0 3.5 4.0 45 t-milliseconds

Fig. 5. Transient response of a conducting sphere in a half space for various depths to the center of the sphere.

The results shown in figs. 3-5 are consistent with these conclusions, provided that the radius of the loop is greater than or equal to the sphere radius. The decay constant for the geometry studied here can readily be found from the respective diagrams. The decay constant of the fields shown in fig. 3 is 6% greater than the decay constant defined above, while the decay constant obtained for the electric field from fig. 5 is 11% too small. Further, the results


also indicate that the decay constant is independent of depth altogether or as a first approximation.

Figs. 6 and 7 indicate the effect of a conducting overburden on the transient response of a buried sphere. Once again the transient curves approach the layered half space response for early times.

Volts/Amp I6 1

F 4.1

0 Varies

d:20m

-T

o=o.I

h-75m

B

0 = 0.01

50m Q = 2.0

(II IOOm IOOD without the sohere (2) IOOm 1006 and Sphere (3) 50~1 loop without the sphere (4) 50m loop with the sphere (5) 30 m loop without the sphere (6) 3Om loop with the sphere (7) 70m loop without the sphere ,,.*-.. I.. .I ,111 ,” m IODP VI,,” me spnere

lo-5 0 0.5 I.0

I I I ! I I I.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

I-milliseconds

Fig. 6. Transient response of a conducting sphere beneath a conducting overburden for various loop sizes.

The effect of the overburden is to shift the departure of the decay curves from the layered ground curves to even later in time. This is dramatically demonstrated by comparing curve number four of fig. 6 with curve number three of fig. 5. Once again the decay constant DC estimated from these decay curves is consistent with decay constant calculated from the formula

DC = +/(&I~).

510 T.LEE

The decay constants for the slowly decaying curves (shown in fig. 7) are 10%

too large. Further, the decay constant is apparently also independent of loop size and

depth of burial. Figs. 6 and 7 show that in this environment it is also necessary to use a very large loop if a reliable estimate of the decay constant is to be obtained from the transient curve from the corresponding background curve.

,ol;Olts/Amp h F4 7,

,631 ’ I I I 0 0.5 I.0 I.5 2.0 2.5 3.0 3.5 4.0 45

t-milliseconds

0

Fig. 7. Transient response of a conducting sphere beneath a conducting overburden for various depths.

ACKNOWLEDGEMENTS

I wish to thank Dr. Roger Lewis for supervising this project and for his help with some of the computer programming.

Professor Keeva Vozoff read the manuscript and made several valuable suggestions for its improvement.


Thanks are also due to Peko Wallsend and L. A. Richardson and Associates for their continuing support.

REFERENCES

ANDREASEN, M. G., rg65,, Scattering from bodies of revolution, Trans. I.E.E.E. 13, 303-310,

BULGAKOV, J. U., 1971, Non Stationary field of an annular current at the surface of a uniform conducting medium. Paper No. 3 in a collection of papers on the transient process method in the search for sulphide ore bodies, “Nedra”, Leningrad.

FULLER, B. D., 1971, The electromagnetic response of a conductive sphere surrounded by a conductive shell, Geophysics 36, g-24.

GARBACZ, R. J. and TURPIN, R. H., 1971, A generalized expansion for radiated and scattered fields, Trans. I.E.E.E. rg, 348-358.

HARRINGTON, R. F., 1968, Field computation by moment methods, New York, zzg. HARRINGTON, R. F. and MAUTZ, J. R., 1971, Theory of characteristic modes for conduct-

ing bodies, Trans. IEEE rg, 622-628. HARRINGTON, R. F. and MAUTZ, J. R., rg7ra, Computation of Characteristic Modes for

Conducting Bodies, Trans. IEEE rg, 629-639. HJELT, S. E., 1971, The transient electromagnetic field of a two layer sphere, Geoexplo-

ration g, 213-230 HOHMANN, G. W., 1971, Electromagnetic scattering by conductors in the earth by a line

source of current, Geophysics 36, 101-131. HOWARD, A. W., 1972, The electromagnetic fields of a subterranean cylindrical in-

homogeneity excited by a line source, Geophysics 37, 975-984. IKEBE, Y., 1972, The Galerkin method for the numerical solution of Fredholm integral

equations of the second kind, SIAM Review 14, 465-491. KAMENETSKI, F. M., 1968, Transient processes using combined loops for a two layer

section with a nonconducting base, Izv. Vozov, sect. Geology Prospecting No. 6. LEE, T. and LEWIS, R. J. G., 1974, Transient EM response of a large loop on a layered

ground, Geophysical Prospecting 22, 430-444. LEE, T., 1974, Transient electromagnetic response of a sphere in a layered medium,

.Unpublished PhD thesis, Macquarie University, North Ryde, NSW Australia. LUKE, Y. L., 1962, Integrals of Bessel functions, McGraw Hill, 4rgp. MACROBERT, T. M., 1967, Spherical harmonics, Pergamon Press, 345~. MAGNUS, W., OBERHETTINGER, F. and SONI, R. P., 1966, Formulas and theorems for the

special functions of mathematical physics, Springer Verlag, 508~. MALLICK, K., 1972, Conducting sphere in an electromagnetic input field, Geophysical

Prospecting 20, 293-304. MORRISON, H. F., PHILLIPS P. J. and O’BRIEN, D. P., 1969, Quantitativeinterpretationof

transient electromagnetic fields over a layered half space, Geophysical Prospecting r7, 82-101.

NABIGHIAN, M. N., 1970, Quasistatic transient response of a conducting sphere in a dipolar field, Geophysics 35, 303-309.

NABIGHIAN, M. N., 1971, Quasistatic response of a conducting permeable two layered sphere in a dipolar field, Geophysics 36, 25-37.

NEGI, J. G. and VERMA, S. K., 1972, Time domain electromagnetic response of a shielded conductor, Geophysical Prospecting 20, gor-gog.

SINGH, R. N., 1972, Transient electromagnetic screening in a two concentric spherical shell model, Pure and Applied Geophysics 94, 226-232.

SINGH, R. N., 1973, Electromagnetic transient response of a conducting sphere in a conductive medium, Geophysics 38, 864-893.

VELEKIN, B. and BULGAKOV, J. U., 1967, Transient method of electrical prospecting (one loop version), International Seminar on Geopysical Methods, Moscow, USSR.

512 LEE, TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE

VELEKIN, A. B., 1971, Transient field of a circular loop in the presence of a conducting sphere. Fourth paper in a collection of papers on the transient processes method in the search for sulphide ore bodies, “Nedra”, Leningrad, 243.

VERMA, S. K. and SINGH, R. N., 1970, Transient electromagnetic response of an inho- mogeneous conducting sphere, Geophysics 35, 331-336.

VERMA, S. K., 1972, Transient Electromagnetic Response of a conducting sphere excited by different types of Input Pulses, Geophysical Prospecting 20, 752-771.

WAIT, J. R. and HILL, D. A., 1972, Electromagnetic surface fields produced by a pulse- excited loop buried in the earth, J. Appl. Physics 43, 3988-3991.

WAIT, J. R., 1972, Transient magnetic fields produced by a step function excited loop buried in the earth, Electronic Letters 8, ooo-ooo.

WATSON, G. N., 1968, A treatise on the theory of Bessel Functions. Second Edition. Cambridge University Press, 804~.

TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE IN A LAYERED MEDIUM

Documents

Transcript of TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE IN A LAYERED MEDIUM